386 XV NORMED ALGEBRAS AND SPECTRAL THEORY be an extremal point of Pj. To prove the converse, show that if /ePi is such that 1 = (l/ll = \f(e)\ then there exists ze A such that \\z\\ < I and /0*z)^0. Using (15.6.11.1), show that the linear forms MX) **f(z*zx)jf(z*z) and MX) =/((« ~~z*z)x)l(f(e) -/(z*z)) belong to Pl5 and that if /is an extremal point then /=/i =/2. Hence we have f(z*zx)=f(x)f(z*z); replacing z by z(e 4- y) where y is small, deduce that /is a character.) (c) Show that for each/e PA there exists a unique positive measure \if on PI, with mass 1 , such that for each x e A we have (Use Section 13.10, Problem 2(b). For the uniqueness of /*/, use the Stone-Weierstrass theorem.) 10. REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS Let K be a compact metrizable space. The application of the results of Section 15.9 to the case where A = ^C(K) allows us to describe very simply all the representations of this algebra. We consider first the topologically cyclic representations u \—> T(u). (15.10.1) Let K be a compact metrizable space and A = ^C(K). Then every topologically cyclic representation of the commutative algebra with involution A, in a separable Hilbert space E, is equivalent to one of the representations u\-^Mti(u) defined as follows: let \JL be a positive measure on K, and for each u e A, let Mp(u) denote the continuous operator on Lc(K, u) induced by multi- plication by u: that is, for each /e J$?c(K, //)> M^u) -f is the class of uf in Let a be a totalizing vector for a representation WH* T(u) of A in a sepa- rable Hilbert space E. The representation Tis determined up to equivalence by the positive linear formfa(u) = (T(u) • a\ a) on A (15.6). Since A is separable (7.4.4), the Bochner-Godement theorem applies; all the characters of A are hermitian, and the spectrum X(A) can be canonically identified with K (15.3.7). Hence the proposition is an immediate consequence of (15.9.2(iv)). It follows from the definition (15.10.1) of M^ that (15.10.2) IIM»|| = ess sup \u(i)\ = N^w) reK